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Axioms of chain complexes

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We find what conditions on chain complexes guarantee the Eilenberg–Steenrod axioms of the homology they produce. Then these axioms become theorems, proven via homological algebra.

In the Eilenberg–Steenrod axioms, the category of all pairs can be replaced with any subcategory ${\mathscr Top}$ as long as it satisfies these conditions (so that we are able to prove theorems we want):

  1. All one-point spaces belong to $\text{Obj}({\mathscr Top})$;
  2. If $(X,A) \in \text{Obj}({\mathscr Top})$ then $(X,X), (X,\emptyset), (A,A), (A,\emptyset) \in \text{Obj}({\mathscr Top})$;
  3. If $(X,A), (Y,D) \in \text{Obj}({\mathscr Top})$ then $(X, A) \times (Y, B) = (X \times Y,A \times Y \cup X \times B) \in \text{Obj}({\mathscr Top})$.

We consider a functor $C$ from ${\mathscr Top}$ to $Ch(\mathcal{Mod})$:

  • $(X,A) \in \text{Obj}({\mathscr Top}) \Rightarrow C(X,A) \in \text{Obj}(Ch(\mathcal{Mod})),$
  • $f \in \text{Hom} ((X,A),(Y,B)) \Rightarrow C(f) \in \text{Hom}(C(X,A),C(Y,B)).$

The five axioms are:

1. Homotopy: Homotopic maps induce chain homotopic chain maps. That is, if two maps are homotopic: $$g \sim h:(X, A) \rightarrow (Y,B),$$ then their induced chain maps are chain homotopic: $$C(g) \simeq C(h):C(X, A) \rightarrow C(Y,B).$$

Holds for singular chain complexes. Implies Homotopy Axiom of Homology.

2. Refinement: Refinement/subdivision doesn't change the chain complex. That is, if $(X,A)$ is a pair and $\gamma$ is an open cover of $X$, then the chain map generated by the inclusion of the sum of subcomplexes $$C(i) : \sum _{U \in \gamma} C(X \cap U, A \cap U) \to C(X, A)$$ is a chain equivalence.

Holds for singular chain complexes. Implies Excision Axiom of Homology, Bredon p. 228.

3. Dimension: The chain complex of the one-point space $P$ is acyclic. That is, the boundary operator $\partial$ of chain complex $C(P)$ of $P$ satisfies $$\text{im } \partial _{k+1} = \ker \partial _{k}, \forall k\ne 0.$$

Trivially holds for cell and singular chain complexes and trivially implies Dimension Axiom of Homology.

4. Additivity: Calculus is additive. That is, if space is the disjoint union of a family of topological spaces $\{X_{\alpha}\}$: $$X = \coprod_{\alpha}{X_{\alpha}},$$ then its chain complex is the direct sum of their chain complexes: $$C(X) \cong \bigoplus_{\alpha} C(X_{\alpha}).$$

Trivially holds for cell and singular chain complexes. Easily implies Additivity Axiom of Homology.

5. Exactness: For any pair $(X,A)$ the sequence $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccccccc} 0& \ra{} & C_m(A) & \ra{C(i)} & C_m(X) & \ra{C(j)} & C_m(X,A) & \ra{} & 0,\\ \end{array} $$ where $i:A \rightarrow X$ and $j:(X,\emptyset ) \rightarrow (X,A)$ are the inclusions, is exact.

Holds for singular chain complexes. Implies Exactness Axiom of Homology, Bredon p. 180.

Note: Since the homology is derived from the chain complex, the latter contains at least as much information about the topology of the space. Taking into account the fact that the cohomology, also derived from the chain complex, may contain, as a ring, more information than the homology (the sphere with two loops vs the torus), we conclude that, in general, the chain complex contains at least as much as and may contain more information about the topology of the space than the homology.